Answer
\begin{align*}
\mathrm{T} &=\frac{6}{7} \mathrm{i}-\frac{2}{7} \mathrm{j}-\frac{3}{7} \mathrm{k}\\
\text{Length } &=49\end{align*}
Work Step by Step
Since
$\mathbf{r}=6 t^{3} \mathbf{i}-2 t^{3} \mathbf{j}-3 t^{3} \mathbf{k}$
Then
$ \mathbf{v}=18 t^{2} \mathbf{i}-6 t^{2} \mathbf{j}-9 t^{2} \mathbf{k} $
and $$|\mathbf{v}|=\sqrt{\left(18 t^{2}\right)^{2}+\left(-6 t^{2}\right)^{2}+\left(-9 t^{2}\right)^{2}}=\sqrt{441 t^{4}}=21 t^{2}$$
Hence
\begin{align*}
\mathrm{T}&=\frac{\mathrm{v}}{\mathrm{|v|}}\\
&=\frac{18 t^{2}}{21 t^{2}} \mathrm{i}-\frac{66^{2}}{21 t^{2}} \mathrm{j}-\frac{9^{2} t^{2}}{21 t^{2}} \mathrm{k}\\
&=\frac{6}{7} \mathrm{i}-\frac{2}{7} \mathrm{j}-\frac{3}{7} \mathrm{k}\\
\text{Length }&=\int_{1}^{2} 2 \mathrm{l} t^{2} d t\\
&=\left[7 t^{3}\right]_{1}^{2}\\
&=49\end{align*}
